DOCSLIB.ORG

Some First-Order Probability Logics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some First-Order Probability Logics Theoretical Computer Science 247 (2000) 191–212 www.elsevier.com/locate/tcs View metadata, citation and similar papers at core.ac.uk brought to you by CORE Some ÿrst-order probability logics provided by Elsevier - Publisher Connector Zoran Ognjanovic a;∗, Miodrag RaÄskovic b a MatematickiÄ institut, Kneza Mihaila 35, 11000 Beograd, Yugoslavia b Prirodno-matematickiÄ fakultet, R. Domanovicaà 12, 34000 Kragujevac, Yugoslavia Received June 1998 Communicated by M. Nivat Abstract We present some ÿrst-order probability logics. The logics allow making statements such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. We describe the corresponding probability models. We give a sound and complete inÿnitary axiomatic system for the most general of our logics, while for some restrictions of this logic we provide ÿnitary axiomatic systems. We study the decidability of our logics. We discuss some of the related papers. c 2000 Elsevier Science B.V. All rights reserved. Keywords: First order logic; Probability; Possible worlds; Completeness 1. Introduction In recent years there is a growing interest in uncertainty reasoning. A part of inves- tigation concerns its formal framework – probability logics [1, 2, 5, 6, 9, 13, 15, 17–20]. Probability languages are obtained by adding probability operators of the form (in our notation) P¿s to classical languages. The probability logics allow making formulas such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. Probability models similar to Kripke models are used to give semantics to the probability formulas so that interpreted formulas are either true or false. Every world from a probability model is equipped with a probability space. The corresponding probability measures are deÿned on sets of subsets of possible worlds. In this paper we present some ÿrst-order probability logics and explore their com- pleteness issues. The most general of our logics (denoted by LFOP1) is similar to the logic with probabilities on possible worlds [1, 9] which is appropriate for ana- lyzing degrees of belief. It is proved in [1] that no complete ÿnitary axiomatization is possible for that logic, and also that even its monadic fragment is undecidable. ∗ Corresponding author. E-mail: [email protected]. 0304-3975/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S0304-3975(98)00341-7 192 Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 We give an inÿnitary axiomatic system for LFOP1 which we prove is sound and complete. In this paper the terms ÿnitary and inÿnitary concern meta language only. Object languages are countable, formulas are ÿnite, while only proofs are allowed to be inÿnite. We also put some restrictions on LFOP1 and investigate the resulting logics. The re- strictions are of the following kinds: only probability measures with ÿxed ÿnite range are allowed in models, only one probability measure on sets of possible worlds is allowed in a model, the measures are allowed to be ÿnitely additive, the set of mea- surable sets of possible worlds can be precisely described, etc. It is interesting that some of the logics obtained by the mentioned constraints have ÿnitary axiomatization, while some fragments are decidable. The rest of the paper is organized as follows. In Section 2 the most general of our logics is introduced, and its syntax, semantics and axiomatization are given. Deÿni- tions, formulations of statements and proofs from this section will be widely used in the remaining sections. In Section 3 we consider the case when only probabilities with ÿxed ÿnite range are allowed and give the corresponding axiomatization. In Section 4 an example which illustrates the relation between probability and modal logics is pre- sented. In Section 5 we consider the case when only one probability measure on sets of possible worlds is allowed. A further restriction of the ÿrst-order probability logics is given in Section 6 where some propositional probability logics are mentioned. In Section 7 the decidability of the considered logics is discussed and a decidable frag- ment of the probability logics is emphasized. We discuss some related papers in Section 8. 2. The logic LFOP1 In this section we present the logic LFOP1 (L for logic, FO for ÿrst-order, and P for probability). We describe its syntax and some classes of models, give an inÿnitary axiomatization and prove that it is sound and complete with respect to the mentioned classes of models. 2.1. Syntax The language L of the LFOP1-logic is an extension of the classical ÿrst-order language. It is a countable set which contains for each nonnegative integer k, k-ary k k k k relation symbols P0 ;P1 ;:::; and k-ary function symbols F0 ;F1 ;:::; and the logical symbols ∧, and ¬, quantiÿer ∀, a list of unary probability operators P¿s for every rational number s ∈ [0; 1], variables x;y;z;:::; and parentheses. The function symbols of the arity 0 are called constant symbols. Terms and atomic formulas are deÿned as in the ÿrst-order classical logic. The set of formulas is the smallest set containing atomic formulas and closed under forma- tion rules: if and ÿ are formulas, then ¬ , P¿s , ∧ ÿ and (∀x) are formulas. Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 193 1 2 0 1 0 For example, the following is a formula: P¿s(∀x)P1 (x) → P3 (y; F0 ) ∧ P¿rP¿tP1 (F1 ). A formula is called classical if it contains no probability operators. In this section ;ÿ; ;::: are used to denote formulas. In a formula of the form (∀x) , is said to be the scope of that quantiÿer. An occurrence of a variable x in a formula is bound if it occurs in a part of which is of the form (∀x)ÿ. Otherwise, the occurrence is called free. A formula is a sentence if no variable is free in .If is a formula and t is a term, then t is said to be free for x in if no free occurrences of x lie in the scope of any quantiÿer (∀y), where y is a variable in t.If is a formula, and x1;:::;xm are variables, (x1;:::;xm) indicates that free variables of form a subset of {x1;:::;xm}.If (x) is a formula, and t is a term free for x in , then (t=x) denotes the result of substituting in the term t for all free occurrences of x. We will also use the shorter form (t) to denote the same substitution. We abbreviate: ¬(¬ ∧¬ÿ)by( ∨ ÿ), (¬ ∨ ÿ)by( → ÿ), ( ↔ ÿ)by(( → ÿ) ∧ (ÿ → )), ¬(∀x)¬ by (∃x) , ¬P¿s( )byP¡s( ), P¿1−s(¬ )byP6s( ), ¬P6s( )by P¿s( ), and P¿s( ) ∧¬P¿s( )byP=s( ). 2.2. Semantics We use the possible-worlds approach to give semantics to probabilistic formulas. It is similar to the objectual interpretation for ÿrst-order modal logics [7]. A class H of subsets of a nonempty set V is an algebra if it contains V and is closed under complementation and ÿnite union. A ÿnitely additive probability measure is a function from an algebra H to the real interval [0; 1] which satisÿes: (V )=1 and (H1 ∪ H2)=(H1)+(H2), for all disjoint sets H1;H2 ∈ H. An algebra H is a -algebra if it is closed under countable union. A function is a probability mea- sure if it maps a -algebra H to the real interval [0; 1] and satisÿes: (V )=1 and S∞ P∞ ( i=1Hi)= i=1 (Hi), for every disjoint sequence {Hi} of sets in H. An LFOP1-model is a structure M = hW; D; I; Probi where: • W is a nonempty set of objects called worlds, • D associates a nonempty domain D(w) with every world w ∈ W , • I associates an interpretation I(w) with every world w ∈ W such that: k k ◦ I(w)(Fi ) is a function from D(w) to D(w), for all i, and k, k k ◦ I(w)(Pi ) is a relation over D(w) , for all i, and k. • Prob is a probability assignment which assigns to every w ∈ W a probability space, such that Prob(w)=hW (w);H(w);(w)i, where: ◦ W (w) is a nonempty subset of W , ◦ H(w) is an algebra of subsets of W (w) and ◦ (w):H(w) → [0; 1] is a ÿnitely additive probability measure. Let M = hW; D; I; Probi be an LFOP1-model. A variable valuation v assigns some element of the corresponding domain to every world w and every variable x, i.e., v(w)(x) ∈ D(w). If w ∈ W , d ∈ D(w), and v is a valuation, then vw[d=x] is a valuation like v except that vw[d=x](w)(x)=d. 194 Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 For a given LFOP1-model M = hW; D; I; Probi, and a valuation v the value of a term t (denoted by I(w)(t)v) is: • if t is a variable x, then I(w)(x)v = v(w)(x), and m • if t = Fi (t1;:::;tm), then m I(w)(t)v = I(w)(Fi )(I(w)(t1)v;:::;I(w)(tm)v). The truth value of a formula in a world w ∈ W for a given LFOP1-model M = hW; D; I; Probi, and a valuation v (denoted by I(w)( )v) is: m m • if = Pi (t1;:::;tm), then I(w)( )v = > if hI(w)(t1)v;:::;I(w)(tm)vi∈I(w)(Pi ), otherwise I(w)( )v = ⊥, • if = ¬ÿ, then I(w)( )v = > if I(w)(ÿ)v = ⊥, otherwise I(w)( )v = ⊥, • if = P¿sÿ, then I(w)( )v = > if (w){u ∈ W (w): I(u)(ÿ)v = >}¿s, otherwise I(w)( )v = ⊥, • if = ÿ ∧ , then I(w)( )v = > if I(w)(ÿ)v = >, and I(w)( )v = >, otherwise I(w)( )v = ⊥, and • if =(∀x)ÿ, then I(w)( ) = > if for every d ∈ D, I(w)(ÿ) = >, otherwise v vw[d=x] I(w)( )v = ⊥.
Load more

Recommended publications
  • Open-Logic-Enderton Rev: C8c9782 (2021-09-28) by OLP/ CC–BY
    An Open Mathematical Introduction to Logic Example OLP Text `ala Enderton Open Logic Project open-logic-enderton rev: c8c9782 (2021-09-28) by OLP/ CC{BY Contents I Propositional Logic7 1 Syntax and Semantics9 1.0 Introduction............................ 9 1.1 Propositional Wffs......................... 10 1.2 Preliminaries............................ 11 1.3 Valuations and Satisfaction.................... 12 1.4 Semantic Notions ......................... 14 Problems ................................. 14 2 Derivation Systems 17 2.0 Introduction............................ 17 2.1 The Sequent Calculus....................... 18 2.2 Natural Deduction......................... 19 2.3 Tableaux.............................. 20 2.4 Axiomatic Derivations ...................... 22 3 Axiomatic Derivations 25 3.0 Rules and Derivations....................... 25 3.1 Axiom and Rules for the Propositional Connectives . 26 3.2 Examples of Derivations ..................... 27 3.3 Proof-Theoretic Notions ..................... 29 3.4 The Deduction Theorem ..................... 30 1 Contents 3.5 Derivability and Consistency................... 32 3.6 Derivability and the Propositional Connectives......... 32 3.7 Soundness ............................. 33 Problems ................................. 34 4 The Completeness Theorem 35 4.0 Introduction............................ 35 4.1 Outline of the Proof........................ 36 4.2 Complete Consistent Sets of Sentences ............. 37 4.3 Lindenbaum's Lemma....................... 38 4.4 Construction of a Model
    [Show full text]
  • Axiomatic Systems for Geometry∗
    Axiomatic Systems for Geometry∗ George Francisy composed 6jan10, adapted 27jan15 1 Basic Concepts An axiomatic system contains a set of primitives and axioms. The primitives are Adaptation to the object names, but the objects they name are left undefined. This is why the primi- current course is in tives are also called undefined terms. The axioms are sentences that make assertions the margins. about the primitives. Such assertions are not provided with any justification, they are neither true nor false. However, every subsequent assertion about the primitives, called a theorem, must be a rigorously logical consequence of the axioms and previ- ously proved theorems. There are also formal definitions in an axiomatic system, but these serve only to simplify things. They establish new object names for complex combinations of primitives and previously defined terms. This kind of definition does not impart any `meaning', not yet, anyway. See Lesson A6 for an elaboration. If, however, a definite meaning is assigned to the primitives of the axiomatic system, called an interpretation, then the theorems become meaningful assertions which might be true or false. If for a given interpretation all the axioms are true, then everything asserted by the theorems also becomes true. Such an interpretation is called a model for the axiomatic system. In common speech, `model' is often used to mean an example of a class of things. In geometry, a model of an axiomatic system is an interpretation of its primitives for ∗From textitPost-Euclidean Geometry: Class Notes and Workbook, UpClose Printing & Copies, Champaign, IL 1995, 2004 yProf. George K.
    [Show full text]
  • Math 310 Class Notes 1: Axioms of Set Theory
    MATH 310 CLASS NOTES 1: AXIOMS OF SET THEORY Intuitively, we think of a set as an organization and an element be- longing to a set as a member of the organization. Accordingly, it also seems intuitively clear that (1) when we collect all objects of a certain kind the collection forms an organization, i.e., forms a set, and, moreover, (2) it is natural that when we collect members of a certain kind of an organization, the collection is a sub-organization, i.e., a subset. Item (1) above was challenged by Bertrand Russell in 1901 if we accept the natural item (2), i.e., we must be careful when we use the word \all": (Russell Paradox) The collection of all sets is not a set! Proof. Suppose the collection of all sets is a set S. Consider the subset U of S that consists of all sets x with the property that each x does not belong to x itself. Now since U is a set, U belongs to S. Let us ask whether U belongs to U itself. Since U is the collection of all sets each of which does not belong to itself, thus if U belongs to U, then as an element of the set U we must have that U does not belong to U itself, a contradiction. So, it must be that U does not belong to U. But then U belongs to U itself by the very definition of U, another contradiction. The absurdity results because we assume S is a set.
    [Show full text]
  • Axiomatic Foundations of Mathematics Ryan Melton Dr
    Axiomatic Foundations of Mathematics Ryan Melton Dr. Clint Richardson, Faculty Advisor Stephen F. Austin State University As Bertrand Russell once said, Gödel's Method Pure mathematics is the subject in which we Consider the expression First, Gödel assigned a unique natural number to do not know what we are talking about, or each of the logical symbols and numbers. 2 + 3 = 5 whether what we are saying is true. Russell’s statement begs from us one major This expression is mathematical; it belongs to the field For example: if the symbol '0' corresponds to we call arithmetic and is composed of basic arithmetic question: the natural number 1, '+' to 2, and '=' to 3, then symbols. '0 = 0' '0 + 0 = 0' What is Mathematics founded on? On the other hand, the sentence and '2 + 3 = 5' is an arithmetical formula. 1 3 1 1 2 1 3 1 so each expression corresponds to a sequence. Axioms and Axiom Systems is metamathematical; it is constructed outside of mathematics and labels the expression above as a Then, for this new sequence x1x2x3…xn of formula in arithmetic. An axiom is a belief taken without proof, and positive integers, we associate a Gödel number thus an axiom system is a set of beliefs as follows: x1 x2 x3 xn taken without proof. enc( x1x2x3...xn ) = 2 3 5 ... pn Since Principia Mathematica was such a bold where the encoding is the product of n factors, Consistent? Complete? leap in the right direction--although proving each of which is found by raising the j-th prime nothing about consistency--several attempts at to the xj power.
    [Show full text]
  • The Entscheidungsproblem and Alan Turing
    The Entscheidungsproblem and Alan Turing Author: Laurel Brodkorb Advisor: Dr. Rachel Epstein Georgia College and State University December 18, 2019 1 Abstract Computability Theory is a branch of mathematics that was developed by Alonzo Church, Kurt G¨odel,and Alan Turing during the 1930s. This paper explores their work to formally define what it means for something to be computable. Most importantly, this paper gives an in-depth look at Turing's 1936 paper, \On Computable Numbers, with an Application to the Entscheidungsproblem." It further explores Turing's life and impact. 2 Historic Background Before 1930, not much was defined about computability theory because it was an unexplored field (Soare 4). From the late 1930s to the early 1940s, mathematicians worked to develop formal definitions of computable functions and sets, and applying those definitions to solve logic problems, but these problems were not new. It was not until the 1800s that mathematicians began to set an axiomatic system of math. This created problems like how to define computability. In the 1800s, Cantor defined what we call today \naive set theory" which was inconsistent. During the height of his mathematical period, Hilbert defended Cantor but suggested an axiomatic system be established, and characterized the Entscheidungsproblem as the \fundamental problem of mathemat- ical logic" (Soare 227). The Entscheidungsproblem was proposed by David Hilbert and Wilhelm Ackerman in 1928. The Entscheidungsproblem, or Decision Problem, states that given all the axioms of math, there is an algorithm that can tell if a proposition is provable. During the 1930s, Alonzo Church, Stephen Kleene, Kurt G¨odel,and Alan Turing worked to formalize how we compute anything from real numbers to sets of functions to solve the Entscheidungsproblem.
    [Show full text]
  • The Axiom of Choice
    THE AXIOM OF CHOICE THOMAS J. JECH State University of New York at Bufalo and The Institute for Advanced Study Princeton, New Jersey 1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK 0 NORTH-HOLLAND PUBLISHING COMPANY - 1973 AN Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Library of Congress Catalog Card Number 73-15535 North-Holland ISBN for the series 0 7204 2200 0 for this volume 0 1204 2215 2 American Elsevier ISBN 0 444 10484 4 Published by: North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - London Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 PRINTED IN THE NETHERLANDS To my parents PREFACE The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathe- matics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject. The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing.
    [Show full text]
  • First-Order Logic in a Nutshell Syntax
    First-Order Logic in a Nutshell 27 numbers is empty, and hence cannot be a member of itself (otherwise, it would not be empty). Now, call a set x good if x is not a member of itself and let C be the col- lection of all sets which are good. Is C, as a set, good or not? If C is good, then C is not a member of itself, but since C contains all sets which are good, C is a member of C, a contradiction. Otherwise, if C is a member of itself, then C must be good, again a contradiction. In order to avoid this paradox we have to exclude the collec- tion C from being a set, but then, we have to give reasons why certain collections are sets and others are not. The axiomatic way to do this is described by Zermelo as follows: Starting with the historically grown Set Theory, one has to search for the principles required for the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to ex- clude all contradictions and, on the other hand, take them sufficiently wide to retain all the features of this theory. The principles, which are called axioms, will tell us how to get new sets from already existing ones. In fact, most of the axioms of Set Theory are constructive to some extent, i.e., they tell us how new sets are constructed from already existing ones and what elements they contain. However, before we state the axioms of Set Theory we would like to introduce informally the formal language in which these axioms will be formulated.
    [Show full text]
  • 2 Axiomatic Systems
    Hawraa Abbas Almurieb Geometry 2 Axiomatic Systems 2.1. Introduction to Axiomatic Systems We need undefined terms for any axiomatic system to build the axioms upon them. Which are basic worlds or things for the system. 2.1.1 Definitions of the system are all other technical terms of the system are ultimately defined by means of the undefined terms. 2.1.2. An axiom is a list of statements dealing with undefined terms and definitions that are chosen to remain unproved. 2.1.3. A theorem is any statement that can be proven using logical deduction from the axioms. 2.1.4. An axiomatic System is a list of axioms and theorems that dealing with undefined terms. Example: Committees Undefined terms: committee, member Axiom 1: Each committee is a set of three members. Axiom 2: Each member is on exactly two committees. Axiom 3: No two members may be together on more than one committee. Axiom 4: There is at least one committee. Chapter Two: Axiomatic Systems 1 | P a g e Hawraa Abbas Almurieb Geometry 2.1.5. A model for an axiomatic system is a way to define the undefined terms so that the axioms are true. Sometimes it is easy to find a model for an axiomatic system, and sometimes it is more difficult. Example: Members Ali, Abbas, Ahmed, Huda, Zainab, Sara. Committees {Ali, Abbas, Ahmed} {Ali, Huda, Zainab} {Abbas, Huda, Sara} {Ahmed, Zainab, Sara} 2.1.6.An axiom is called independent if it cannot be proven from the other axioms. Example: Consider Axiom 1 from the Committee system.
    [Show full text]
  • Randomness in Arithmetic and the Decline and Fall of Reductionism in Pure Mathematics
    RANDOMNESS IN ARITHMETIC AND THE DECLINE AND FALL OF REDUCTIONISM IN PURE MATHEMATICS IBM Research Report RC-18532 November 1992 arXiv:chao-dyn/9304002v1 7 Apr 1993 G. J. Chaitin Lecture given Thursday 22 October 1992 at a Mathematics – Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript. 1. Hilbert on the axiomatic method Last month I was a speaker at a symposium on reductionism at Cam- bridge University where Turing did his work. I’d like to repeat the talk 1 2 I gave there and explain how my work continues and extends Turing’s. Two previous speakers had said bad things about David Hilbert. So I started by saying that in spite of what you might have heard in some of the previous lectures, Hilbert was not a twit! Hilbert’s idea is the culmination of two thousand years of math- ematical tradition going back to Euclid’s axiomatic treatment of ge- ometry, going back to Leibniz’s dream of a symbolic logic and Russell and Whitehead’s monumental Principia Mathematica. Hilbert’s dream was to once and for all clarify the methods of mathematical reasoning. Hilbert wanted to formulate a formal axiomatic system which would encompass all of mathematics. Formal Axiomatic System −→ −→ −→ Hilbert emphasized a number of key properties that such a formal axiomatic system should have. It’s like a computer programming lan- guage. It’s a precise statement about the methods of reasoning, the postulates and the methods of inference that we accept as mathemati- cians.
    [Show full text]
  • Axiomatic Systems
    Eucliean and Non-Euclidean Geometry – Fall 2007 Dr. Hamblin Axiomatic Systems An axiomatic system is a list of undefined terms together with a list of statements (called “axioms”) that are presupposed to be “true.” A theorem is any statement that can be proven using logical deduction from the axioms. Examples Here are some examples of axiomatic systems. Committees Undefined terms: committee, member Axiom 1: Each committee is a set of three members. Axiom 2: Each member is on exactly two committees. Axiom 3: No two members may be together on more than one committee. Axiom 4: There is at least one committee. Monoid Undefined terms: element, product of two elements Axiom 1: Given two elements x and y, the product of and , denoted , is a uniquely defined element. Axiom 2: Given elements , , and , the equation is always true. Axiom 3: There is an element , called the identity, such that and for all elements . Silliness Undefined terms: silly, dilly. Axiom 1: Each silly is a set of exactly three dillies. Axiom 2: There are exactly four dillies. Axiom 3: Each dilly is contained in a silly. Axiom 4: No dilly is contained in more than one silly. Notice that in the second example, the axioms defined a new term (“identity”). This isn’t an undefined term because the axiom includes a definition. Also, these axioms refer to basic set theory that you learned in Discrete Math. For our purposes, we will assume all of those basic set theory terms are known. It is possible to view set theory itself as another axiomatic system, but that is beyond the scope of this course.
    [Show full text]
  • First-Order Logic in a Nutshell
    Chapter 2 First-Order Logic in a Nutshell Mathematicians devised signs, not separate from matter except in essence, yet distant from it. These were points, lines, planes, solids, numbers, and countless other characters, which are depicted on paper with certain colours, and they used these in place of the things sym- bolised. GIOSEFFO ZARLINO Le Istitutioni Harmoniche, 1558 First-Order Logic is the system of Symbolic Logic concerned not only with rep- resenting the logical relations between sentences or propositions as wholes (like Propositional Logic), but also with their internal structure in terms of subject and predicate. First-Order Logic can be considered as a kind of language which is dis- tinguished from higher-order languages in that it does not allow quantification over subsets of the domain of discourse or other objects of higher type. Nevertheless, First-Order Logic is strong enough to formalise all of Set Theory and thereby vir- tually all of Mathematics. In other words, First-Order Logic is an abstract language that in one particular case might be the language of Group Theory, and in another case might be the language of Set Theory. The goal of this brief introductionto First-Order Logic is to illustrate and summarise some of the basic concepts of this language and to show how it is applied to fields like Group Theory and Peano Arithmetic (two theories which will accompany us for a while). Syntax: The Grammar of Symbols Like any other written language, First-Order Logic is based on an alphabet, which consists of the following symbols: 11 12 2 First-Order Logic in a Nutshell (a) Variables such as v0, v1, x, y, .
    [Show full text]
  • The Axiomatic Method in Sociology
    Sarah Wylie HON 213 4/25/05 The Axiomatic Method in Sociology The written works of social theorists can be seen as examples of axiomatic method arguments in the world of social science. In fact, the method of rational proof of statements which has been engrained in our society since the Enlightenment can be seen throughout the academic disciplines. This paper will briefly outline some of the ways in which mathematicians and sociologists employ the axiomatic method in similar and different ways. The axiomatic method is a way of proving the truth of a certain statement. The statement is said to be true if it follows logically from a number of statements that are in turn the logical result of other statements. At the very beginning of a proof by the axiomatic method there are certain special statements called axioms that are taken as true without justification. In order to make arguments by the axiomatic method, there must exist something that both the writer and audience agree upon to take as true without any particular justification. If there are no statements that the arguer and audience can both agree on, or at the very least accept for the sake of argument, then there is no way that the theorist can further his argument in the axiomatic method. In math, axioms are generally basic statements about the elements of a system. For example, the first incidence axiom of geometry is a statement about lines and points, “for every two distinct points there exists a unique line incident on them.” Similarly, what could be considered axioms in sociology are basic statements about the elements of sociology.
    [Show full text]